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Many engineering, social, and biological complex systems consist of dynamicalelements connected via a large-scale network. Monitoring the network's dynamicsis essential for a variety of maintenance and scientific purposes. Whilst weunderstand how to optimally sample a single dynamic element or a non-dynamicgraph, we do not possess a theory on how to optimally sample networkeddynamical elements. Here, we study nonlinear dynamic graph signals on a fixedcomplex network. We define the necessary conditions for optimal sampling in thecombining time- and graph-domain to fully recover the networked dynamics. Wefirstly interpret the networked dynamics into a linearized matrix. Then, weprove that the dynamic signals can be sampled and fully recovered if thenetworked dynamics is stable and their initialization is bandlimited in thegraph spectral domain. This new theory directly maps optimal sampling locationsand rates to the graph properties and governing nonlinear dynamics. This caninform the optimal placement of experimental probes and sensors on dynamicalnetworks as well as inform the design of each sensor's optimal sampling rate.We motivate the reader with two examples of recovering the networked dynamicsfor: social population growth and networked protein biochemical interactionswith both bandlimited and arbitrary initialization.

Optimal Sampling for Dynamic Complex Networks with Graph-Bandlimited Initialization